Download Other Number Systems & Base-R to Decimal

Number Systems
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Information Representation
 Numbers are important to computers
 represent information precisely
 can be processed
 For example:
 to represent yes or no: use 0 for no and 1 for yes
 to represent 4 seasons: 0 (autumn), 1 (winter), 2(spring)
and 3 (summer)
 roll number (8 alphanumeric) to represent individual
students
Information Representation
 Elementary storage units inside computer are electronic
switches. Each switch holds one of two states: on (1) or
off (0).
ON
OFF
 We use a bit (binary digit), 0 or 1, to represent the state.
Information Representation
 Storage units can be grouped together to cater for larger
range of numbers.
 Example: 2 switches to represent 4 values.
0 (00)
1 (01)
2 (10)
3 (11)
Information Representation
 In general, N bits can represent 2N different values.
 For M values, log 2 M  bits are needed.
1 bit  represents up to 2 values (0 or 1)
2 bits  rep. up to 4 values (00, 01, 10 or 11)
3 bits  rep. up to 8 values (000, 001, 010. …, 110, 111)
4 bits  rep. up to 16 values (0000, 0001, 0010, …, 1111)
32 values  requires 5 bits
64 values  requires 6 bits
1024 values  requires 10 bits
40 values  requires 6 bits
100 values  requires 7 bits
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Positional Notations
 Position-independent notation
 each symbol denotes a value independent of its position:
Egyptian number system
 Relative-position notation
 Roman numerals symbols with different values: I (1), V (5),
X (10), C (50), M (100)
 Examples: I, II, III, IV, VI, VI, VII, VIII, IX
 Relative position important: IV = 4 but VI = 6
 Computations are difficult with the above two notations
Positional Notations
 Weighted-positional notation
 Decimal number system, symbols = { 0, 1, 2, 3, …, 9 }
 Position is important
 Example:(7594)10 = (7x103) + (5x102) + (9x101) + (4x100)
 The value of each symbol is dependent on its type and its
position in the number
 In general, (anan-1… a0)10 = (an x 10n) + (an-1 x 10n-1) + …
+ (a0 x 100)
Positional Notations
 Fractions are written in decimal numbers after the decimal point.
 2
3
4
= (2.75)10 = (2 x 100) + (7 x 10-1) + (5 x 10-2)
 In general, (anan-1… a0 . f1f2 … fm)10 = (an x 10n) + (an-1x10n-1)
+ … + (a0 x 100) + (f1 x 10-1) + (f2 x 10-2) + … + (fm x 10-m)
 The radix (or base) of the number system is the total number of
digits allowed in the system.
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Decimal (base 10) Number System
 Weighting factors (or weights) are in powers-of-10:
… 103 102 101 100.10-1 10-2 10-3 10-4 …
 To evaluate the decimal number 593.68, the digit in each position
is multiplied by the corresponding weight:
5102 + 9101 + 3100 + 610-1 + 810-2
= (593.68)10
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Other Number Systems & Base-R to Decimal Conversion
 Binary (base 2): weights in powers-of-2.
 Binary digits (bits): 0,1.
 Octal (base 8): weights in powers-of-8.
 Octal digits: 0,1,2,3,4,5,6,7.
 Hexadecimal (base 16): weights in powers-of-16.
 Hexadecimal digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
 Base R: weights in powers-of-R.
Other Number Systems & Base-R to Decimal Conversion
 (1101.101)2 = 123 + 122 + 120 + 12-1 + 12-3
= 8 + 4 + 1 + 0.5 + 0.125 = (13.625)10
 (572.6)8 = 582 + 781 + 280 + 68-1
= 320 + 56 + 2 + 0.75 = (378.75)10
 (2A.8)16 = 2161 + 10160 + 816-1
= 32 + 10 + 0.5 = (42.5)10
 (341.24)5 = 352 + 451 + 150 + 25-1 + 45-2
= 75 + 20 + 1 + 0.4 + 0.16 = (96.56)10
Other Number Systems & Base-R to Decimal Conversion
 Counting in Binary
 Assuming non-negative values,
n bits  largest value 2n – 1.
Examples: 4 bits  0 to 15;
6 bits  0 to 63.
 Range of m values  log2m bits
Decimal
Number
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Binary
Number
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Decimal-to-Binary Conversion
 Method 1: Sum-of-Weights Method
 Method 2:
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
Sum-of-Weights Method
 Determine the set of binary weights whose sum is equal to
the decimal number.
(9)10 = 8 + 1 = 23 + 20 = (1001)2
(18)10 = 16 + 2 = 24 + 21 = (10010)2
(58)10 = 32 + 16 + 8 + 2 = 25 + 24 + 23 + 21 = (111010)2
(0.625)10 = 0.5 + 0.125 = 2-1 + 2-3 = (0.101)2
Repeated Division-by-2 Method
 To convert a whole number to binary, use successive division
by 2 until the quotient is 0. The remainders form the answer,
with the first remainder as the least significant bit (LSB) and
the last as the most significant bit (MSB).
(43)10 = (101011)2
2
2
2
2
2
2
43
21
10
5
2
1
0
rem 1  LSB
rem 1
rem 0
rem 1
rem 0
rem 1  MSB
Repeated Multiplication-by-2 Method
 To convert decimal fractions to binary, repeated multiplication by
2 is used, until the fractional product is 0 (or until the desired
number of decimal places). The carried digits, or carries,
produce the answer, with the first carry as the MSB, and the last
as the LSB.
(0.3125)10 = (.0101)2
0.31252=0.625
0.6252=1.25
0.252=0.50
0.52=1.00
Carry
0
1
0
1
MSB
LSB
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Conversion between Decimal and other Bases
 Base-R to decimal: multiply digits with their corresponding
weights.
 Decimal to binary (base 2)
 whole numbers: repeated division-by-2
 fractions: repeated multiplication-by-2
 Decimal to base-R
 whole numbers: repeated division-by-R
 fractions: repeated multiplication-by-R
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Conversion between Bases
 In general, conversion between bases can be done via decimal:
Base-2
Base-3
Base-4
…
Base-R
Decimal
Base-2
Base-3
Base-4
….
Base-R
 Shortcuts for conversion between bases 2, 4, 8, 16.
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Binary-Octal/Hexadecimal Conversion
 Binary  Octal: Partition in groups of 3
(10 111 011 001 . 101 110)2 = (2731.56)8
 Octal  Binary: reverse
(2731.56)8 = (10 111 011 001 . 101 110)2
 Binary  Hexadecimal: Partition in groups of 4
(101 1101 1001 . 1011 1000)2 = (5D9.B8)16
 Hexadecimal  Binary: reverse
(5D9.B8)16 = (101 1101 1001 . 1011 1000)2
Outline
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Information Representations
Positional Notations
Decimal (base 10) Number System
Other Number Systems & Base-R to Decimal Conversion
Decimal-to-Binary Conversion
 Sum-of-Weights Method
 Repeated Division-by-2 Method (for whole numbers)
 Repeated Multiplication-by-2 Method (for fractions)
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Conversion between Decimal and other Bases
Conversion between Bases
Binary-Octal/Hexadecimal Conversion
Binary Arithmetic Operations
Binary Arithmetic Operations
 ADDITION
 Like decimal numbers, two numbers can be added by adding each
pair of digits together with carry propagation.
(11011)2
+ (10011)2
(101110)2
(647)10
+ (537)10
(1184)10
Binary Arithmetic Operations
 Digit addition table:
BINARY
DECIMAL
0+0+0=00
0+0+0=00
0+1+0=01
0+1+0=01
1+0+0=01
1+1+0=10
0+2+0=02
…
0+0+1=01
1+8+0=09
0+1+1=10
1+9+0=10
1+0+1=10
…
1+1+1=11
9+9+1=19
Carry in
Carry out
(11011)2
+ (10011)2
(101110)2
1 0
0 1
0 1
1 0
0 1
0 1 1 0
1 0 1 1
0 0 1 1
1 1 1 0
0 0 1 1
Binary Arithmetic Operations
 SUBTRACTION
 Two numbers can be subtracted by subtracting each pair of
digits together with borrowing, where needed.
(11001)2
- (10011)2
(00110)2
(627)10
- (537)10
(090)10
Binary Arithmetic Operations
 Digit subtraction table:
BINARY
DECIMAL
0-0-0=00
0-0-0=00
0-1-0=11
0-1-0=19
1-0-0=01
0-2-0=18
1-1-0=00
…
0-0-1=11
0-9-1=10
0-1-1=10
1-0-1=00
1-0-1=00
…
1-1-1=11
9-9-1=19
Borrow
(11001)2
- (10011)2
(00110)2
0 0 1
0 1 1
0 1 0
0 0 0
0 0 0
1 0 0
0 0 1
0 1 1
1 1 0
1 1 0
Binary Arithmetic Operations
 MULTIPLICATION
 To multiply two numbers, take each digit of the multiplier and
multiply it with the multiplicand. This produces a number of
partial products which are then added.
(11001)2
x (10101)2
(214)10
x (152)10
(11001)2
(11001)2
+(11001)2
(1000001101)2
(428)10
(1070)10
+(214)10
(32528)10
Multiplicand
Multiplier
Partial
products
Result
Binary Arithmetic Operations
 Digit multiplication table:
BINARY
DECIMAL
0X0=0
0 X 0= 0
0 X 1= 0
0 X 1= 0
1X0=0
…
1 X 1= 1
1X8=8
1 X 9= 9
…
9 X 8 = 72
9 X 9 = 81
 DIVISION – can you figure out how this is done?
 Think of the division technique (shift & subtract) used for decimal
numbers and apply it to binary numbers.